Degree of Polynomial

The degree of a polynomial is defined as the highest power of the variable in the polynomial expression.

Example: 5x³ + 2x² + 7x + 1

• Highest power of x = 3
• Degree = 3

Degree of Polynomial with more than One Variable

If a polynomial has more than one variable, then its degree is calculated by adding the exponents of each variable.

Degree of Zero Polynomial

A polynomial is said to be a zero polynomial if the coefficients of all the variables are equal to zero. Let the zero polynomial be f (x) = 0. Now, we can write it as f(x) = 0x⁰, f(x) = 0x¹, f(x) = 0x², f(x) = 0x³, etc. So, we can say that the degree of a zero polynomial is undefined. Sometimes it is defined as negative (-1 or -∞).

Degree of Constant Polynomial

A polynomial is said to be a constant polynomial if its value remains the same. Let the constant polynomial be P(x) = c, or we can write it as P(x) = Cx⁰ since the value of x₀ is 1. So, we can say that the degree of constant polynomial is zero.

Example: P(x) = 13 = 13x⁰.

So, the degree of P(x) is zero.

Classification of Polynomials Based on its Degree

The following are some polynomial expressions depending on the degree of a polynomial, with examples.

How to find the degree of a polynomial?

The following are the steps to determine the degree of a polynomial expression. Now, let us find the degree of the polynomial expression 7x⁴+6x³-2x⁴+12x²+9x+3

Step 1: Combine all the like terms of the given polynomial expression, where like terms are the terms that have the same variables and powers. Here, 7x4 and -2x4 are like terms:

(7x⁴ - 2x⁴) + 6x³ + 12x² + 9x + 3 = 5x⁴ + 6x³ + 12x² + 9x + 3.

Step 2: Ignore the coefficients of all the variables.

x⁴ + x³ + x² + x¹ + x⁰

Step 3: Now arrange all the variables in the descending order of their powers, i.e., from the greatest exponent to the least.

x⁴ + x³ + x² + x¹ + x⁰

Step 4: Now identify the largest power of the variable x, as the degree of a polynomial expression is the highest exponent of the variable.

So, the degree of the given polynomial expression is 4.

Related Articles

• How to Find the Degree of a Polynomial with More Than One Variable?
• Polynomial Functions
• Types of Polynomials
• Polynomial Formula
• Implementation of Polynomial Regression

Degree of polynomial function examples

Example 1: Determine the degree, constant, and leading coefficient of the polynomial expression 7x − 8x + 2x + 5.

Solution:

Given Polynomial Expression = 7x⁴ − 8x³ + 2x + 5

The highest exponent of variable x = 4

So, the degree of the given polynomial expression = 4

The leading coefficient of the polynomial is the coefficient with the highest exponent. 

So, the leading coefficient of given the polynomial expression = 7

Constant = 5

Example 2: Determine the degree of the polynomial expression 2x + 6x − x + 3x + x + 9.

Solution:

Given polynomial expression = 2x⁴ + 6x⁵ − x³ + 3x² + x⁶ + 9

The polynomials are not arranged from greatest exponent to least. So, let us arrange them in descending order of their exponents first.

So, the obtained expression = x⁶ + 6x⁵ + 2x⁴ − x³ + 3x² + 9

Now, the highest exponent of the variable x = 6

So, the degree of the given polynomial expression = 6.

Example 3: Find the degree and constant of the polynomial expression 3x − 16x + 21x − 7x.

Solution:

Given polynomial expression = 3x⁸ − 16x⁵ + 21x² − 7x

The highest exponent of the variable x = 8

So, the degree of the given polynomial expression = 8

The constant of the given polynomial expression = 0

Example 4: Determine the degree, constant, and leading coefficient of the polynomial expression 13x − 15x − 11x + 9.

Solution:

Given polynomial expression = 13x³ − 15x² − 11x + 9

The highest exponent of the variable x = 3

So, the degree of the given polynomial expression = 3

The leading coefficient of the polynomial is the coefficient with the highest exponent.

So, the leading coefficient of given the polynomial expression = 13

Constant = 9.

Example 5: Calculate the degree of polynomial 4x³ + 7x³y¹ + 11x²y³ + 17xy + 21y³.

Solution:

The given polynomial expression is 4x³ + 7x³y¹ + 11x²y³ + 17xy² + 21y³.

Now, let's calculate the degree of each term.

4x³ has a degree of 3 since the power of x is 3.

7x³y¹ has a degree of 4 since the power of x is 3 and the power of y is 1. So, by adding the exponents of x and y, we get 4.

11x²y³ has a degree of 5 since the power of x is 2 and the power of y is 3. So, by adding the exponents of x and y, we get 5.

17xy² has a degree of 3 since the power of x is 1 and the power of y is 2. So, by adding the exponents of x and y, we get 3.

21y³ has a degree of 3 since the power of y is 3.

The largest degree out of these is 5, so the degree of the given polynomial expression is 5.

Example 6: Calculate the degree of the polynomial 13x⁴ + 8xy + 7xy² + 11xy.

Solution:

The given polynomial expression is 13x⁴ + 8x³y² + 7x²y+11xy.

Now, let's calculate the degree of each term.

13x⁴ has a degree of 4 since the power of x is 3.

8x³y² has a degree of 5 since the power of x is 3 and the power of y is 3. So, by adding the exponents of x and y, we get 5. 

7x²y has a degree of 3 since the power of x is 2 and the power of y is 1. So, by adding the exponents of x and y, we get 3. 

11xy has a degree of 2 since the power of both x and y is 1. So, by adding the exponents of x and y, we get 2. 

The largest degree out of these is 5, so the degree of the given polynomial expression is 5.